A384419 Exponentially squarefree prime powers.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset

1,2

Comments

Differs from A283262 by having the terms 1, 128, 1024, 2187, 8192, ..., and not having the terms 256, 512, 6561, ... .

1 and numbers of the form p^e where p is prime and e is a squarefree number.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 24.

Mathematica

seq[lim_] := Module[{s = Select[Range[lim], !CompositeQ[#] &]}, Do[If[SquareFreeQ[i], s = Join[s, Select[Range[Surd[lim, i]], PrimeQ]^i]], {i, 2, Floor[Log2[lim]]}]; Sort[s]]; seq[250]

Prog

(Python)
from math import isqrt
from sympy import mobius, integer_log, primerange, primepi
def A384419(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid: kmax = kmid
            else: kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n-1+x-sum(g(integer_log(x, p)[0])-1 for p in primerange(isqrt(x)+1))-primepi(x)
    return bisection(f, n, n) # Chai Wah Wu, Sep 21 2025

Crossrefs

Intersection of A000961 and A209061.

Disjoint union of {1}, A000040 and A383211.

Cf. A005117, A283262.

Sequence in context: A056781 A079446 A322546 * A283262 A334393 A115975

Adjacent sequences: A384416 A384417 A384418 * A384420 A384421 A384422

Keyword

nonn,easy

Author

Amiram Eldar, May 28 2025

Status

approved